A multidimensional generalization of the Erdos-Szekeres lemma on monotone subsequences

We consider an extension of the Monotone Subsequence lemma of Erdos and Szekeres in higher dimensions. Let v(1),..., v(n) is an element of IRd be a sequence of real vectors. For a subset I subset of or equal to [n] and vector (c) over right arrow is an element of {0,1}(d) we say that I is (c) over right arrow -free if there are no i < j is an element of I, such that, for every k = 1,...,d, v(k)(i) < v(k)(j) if and only if (c) over right arrow (k) = 0. We construct sequences of vectors with the k k property that the largest (c) over right arrow -free subset is small for every choice of (c) over right arrow. In particular, for d = 2 the largest (c) over right arrow free subset is O(n(5/8)) for all the four possible (c) over right arrow. The smallest possible value remains far from being determined. We also consider and resolve a simpler variant of the problem.